0

I found this problem in the Sheldon Ross book:

In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is

(a) the probability that a randomly selected family owns both a dog and a cat?

(b) the conditional probability that a randomly selected family owns a dog given that it owns a cat?

The answer to a) is given as $$P(D\cap C) = P(C\vert D)P(D)$$

This is very non-intuitive to me as I can't understand why $$P(D\cap C) \not = P(C \vert D) \not = P(D \vert C)$$

nordrhein
  • 1

1 Answers1

1

You are someone who always bring an umbrella whenever it rains, also, sometimes you might bring an umbrella even when it is not raining.

Let $D$ be the event that it rains, and $C$ is the event that you bring an umbrella.

$$P(C|D)=1$$

However, it doesn't means that it rains all the time, $P(C \cap D)<1$.

Also, it is unlikely that whenever you bring an umbrella, it brings rain. $P(D|C)<1$.

Siong Thye Goh
  • 149,520
  • 20
  • 88
  • 149